1. Thermodynamics II I.Ensembles II.Distributions III. Partition Functions IV. Using partition functions V. A bit on gibbes

2. Ensembles Formally, an ensemble is virtual construct of many copies of a system of interest. Each member of an ensemble has some mechanic or thermodynamic variables fixed, but all states corresponding to these fixed variables all allowed. Suppose we had a single protein molecule in a box of water molecules of fixed volume at a single temperature. Then an ensemble is a collection of these molecules This is why we often call a collection of proteins in a test-tube an ensemble, even though it is not be formally correct. Each state is represented equally in an ensemble; or alternatively an isolated system is equally likely to be found in any one possible quantum state

3. Ensembles Depending on which variables are fixed, we have different ensembles: Practical Example of the microcanoical? If every member of the ensemble has a fixed volume, number of particles, and Temperature, then we are in the canonical ensemble If every member of the ensemble has a fixed volume, number of particles, and E: then we are in the microcanonical ensemble We will be simulating members of the canonical ensemble

4. Other Ensembles Depending on which variables are fixed, we have different ensembles: This is used when simulating systems with variable number of particles e.g. particles absorbed on a surface If every member of the ensemble has a fixed pressure, number of particles, and Temperature, then we are in the isothermal-isobaric ensemble Grand canonical has fixed V and T, and can be constructed by considering copies of canonical ensembles connected by permeable membranes The isothermal-isobaric ensemble is another common one to simulate

5. Ensembles and Reality If a system has a large enough number of molecules, then the behavior of the system will coincide with that predicted from a statistical consideration of the individual molecules. If we wait long enough, a single particle will sample all possible states. Theory and experiment both rely on different aspects of the erogodic hypothesis These are the erogodic hypothesis.

6. Dice and Distributions Suppose we roll two dice: What is the most probable number ? What are the least probable numbers? What are the odds of getting the least probable numbers? Suppose we roll twenty dice: What is the most probable number ? What are the least probable numbers? What are the odds of getting the least probable numbers? So as we increase the number of identical particles, the probability of seeing extreme events decreases. The most probable state dominates, and the range of fluctuations decrease. When the number of members of the ensemble become very large, all macroscopic quanitites are essentially fixed!

7. Ensembles and Distributions We when have an ensemble, there are fixed quantities, and all states which meet these constraints are allowed, and are equally probable for any particle. The most probable distribution is that corresponding to the largest number of ways of arranging particles in a given configuration The number of ways of arranging particles is the statistical weight; for a constant energy and particle number this is W or  Does W look familiar ? Note: the particles need not be molecules; they can be atoms inside molecules. All the force-fields terms discussed last time have entropic contributions

8. Boltzmann Distribution The Boltzmann distribution is the most probable distribution for a large system near equilibrium Notes: This can be proven; those of you in physics/physical chemistry probably will do so or have done so already. This is also the high T limit of the fermi-dirac and Bose-Einstein distributions

9. Distribution and Conformations Experiment: the thermodynamics of HEWL and a mutant missing a disulfide bond were studied with scanning microcalorimetry. Both proteins have the same enthaply of unfolding, and had “two-state” behavior, but there is a difference in the entropy of unfolding. Where does the entropy change come from? The x-ray structures show essentially the same structures and interactions.

10. Distribution and Conformations There are 129 residues in the protein, and if we calculate per residue: Assume the folded states have the same entropy What does this mean? What if we hadn’t divide by 129?

11. Probability and the partition function Using the Boltzmann distribution, the probability at a given temperature of a particular state is: The bottom is the canonical partition function.

12. Ensemble averages We can write macroscopic variables in terms of the parition function Energy, pressure, entropy, helmholz free energy, and more

13. Utility of Other Ensembles: Microcanonical: Isothermal/Isobaric: